Creating a new vert with bmesh resets previously created verts
Add vert at cursor in edit mode

The cursor is in global coordinates. Set to local by multiplying by the objects world matrix inverse.

import bpy
import bmesh

context = bpy.context
scene = context.scene
ob = context.object
mwi = ob.matrix_world.inverted()
me = ob.data
bm = bmesh.from_edit_mesh(me)
bm.verts.new(mwi @ scene.cursor.location)
bmesh.update_edit_mesh(me)